 Thank you, Rick. My thanks to the organizers for inviting me here today to speak to you and for inspiring me to keep the themes of this conference, these conferences, in mind while doing my research. So as Rick said, I want to talk to you about nonlinear interactions of what are called kink unstable flux ropes and shear off in waves in plasmids. And in the end, I hope to show you that from these large scale, generally large scale, structures we can produce at least one step down in creating smaller scale structures, not a cascade, but at least a step towards that. And so I'd like to thank my collaborators, Professor Walter Gekleman who is here with us today, Shikrisha Jatapathi, one of our greatest students, Tim DeHass and Patrick Pribble. And thanks to our funding agencies, the National Science Foundation and the offices of Fusion Energy Science at the Department of Energy. So I'll go into a little bit about the motivations for this research. I'll do a quick review on what I think some key points are the kink oscillations for the plasma, what a shear off in wave is because I know this is a diverse group. Then the experimental setup, I will show you the frequency free wave matching conditions for the interaction of these two different oscillations, k-theta free wave matching conditions satisfying satisfaction. Then the energy transfer from the larger to the smaller scales, so I'll show that quantitatively. And then I will do a biospectral analysis to demonstrate that the interaction is a quadratic, or at least in part is a quadratic energy transfer between the two scales in the summary. So some of the motivations for this are the ubiquity of what are called magnetic flux ropes and shear off in waves in magnetized plasmas. So you find these structures anywhere from the solar surface out into the solar wind and into the magnetosphere, the magnetotail of the Earth and of other planets which have magnetospheres. So these structures can be very large. You can see an example from the trace satellite. This is a magnetic flux rope, which is a bundle of twisted magnetic field lines in a plasma, I'll get into that a little bit more, which is probably larger than the size of the Earth, which then undergoes kink oscillation, causing it to twist a little bit. And this may be a trigger condition for a process called magnetic reconnection, which then allows this process to detach itself from the magnetic fields of the sun and then flow as a coronal mass ejection into the solar wind. Let's see. So very quick review. So imagine you have a plasma, which is carrying a current that has a length L. So plasma here's in this orange color, the magnetic field produced by the current and it's all embedded in the background magnetic field. So the magnetic field produces a current which is going following these arrows as musically around the magnetic flux tube. Plasma has a radius A, this length L. Now suppose you would introduce a small perturbation displacement for the plasma. Let's say it doesn't like snakes very much. So this plasma moves out of the way. You've created a small disturbance in the plasma there. So what does that tend to do? It creates a decreased magnetic pressure on one side and increased magnetic pressure on the other. And you can see that this increase in magnetic pressure then tends to reinforce this perturbation. Now the actual kink instability has a lot more detail than this. But for this audience, I thought this is just a brief introduction to what it might be. So again, these are the dimensions of the plasma. The criterion for this becoming kink unstable is suppose this perturbation can propagate down to some boundaries. So the lowest mode you can have would be a half a wavelength. And can that perturbation, which exists in the cylindrical form in a two-dimensional sense, can it, does the magnetic field cause by this spiral around once by the time it gets all the way down the cylinder? And this leads to this, what's called the Couscous-Chefranoff limit, which I'll show you where we satisfy this and where we don't in the experiment later. But that's just the basic idea. Now for alphane waves, what are they? They are, as I said, like they are ubiquitous phenomenon in magnetized plasmas. They are responsible. They're very low frequency. They propagate below the ion cyclotron frequency. They're responsible for transferring, propagating changes in magnetic field topology at these low frequencies throughout the, for example, the magnetosphere. You find them anywhere from tokamak plasmas, other laboratory plasmas. In the Earth's magnetosphere, for example, they have both perpendicular and parallel electric fields. They can therefore accelerate particles along the back right magnetic field. There are certain types of aurora that are generated by the parallel electric fields carried by alphane waves which are generated in the magnetotail through this process called magnetic reconnection. I mentioned earlier, they accelerate electrons towards both of the poles. They generally have length scales along the background magnetic field, scaled by the ion inertial length, the C over a mega PI, where a mega PI is the plasma frequency. And across the magnetic field, they can have scales down to either the electron or inertial length or the ion gyro radius or what's called the ion sound gyro radius. But the key takeaways is they have both parallel and perpendicular fields and they can be, depending on the frequency, also a very long wavelength. Okay, now, these experiments are performed at the large plasma device, which is the main operational device of the basic plasma science facility is a URL here. I'm encouraging all of you to visit. This is a user facility for basic plasma physics research. It's not just in the US, but participants can come from around the world. Anybody here can propose to do an experiment, even if you're a theorist, ask our chairman. So please, if you have an idea for an experiment, we would like to work with you to bring that to fruition. Either come talk to myself or Walter Gekleman who's up front here. We'll be here all week and we'll be happy to talk to you about it. About doing possible experiments. Okay, the device itself is 20 meters long. So in terms of that length scale, I mentioned the ion inertial length that maybe in this case, we're using a proton plasma. So the ion inertial length, the size of the machine is about 200 ion inertial lengths in length. Across the field, the machine can be, can be highly magnetized or going to lower magnetic fields, but it's a magnetized plasma, some 100 to 1000 ion gyro-radi across the plasma column or up to six ion inertial lengths across the plasma column. There are two sources in the large plasma device. One is a large sheet of nickel, which is heated and coated with a special material, barium oxide, to reduce the work function. This produces a 60 centimeter diameter plasma for the 20 length, 20 meter length of the device. There's a second cathode with higher emissivity. It allows the creation of a hotter denser plasma within this ambient background plasma. So we use that in conjunction with a mask, which is just a piece of carbon graphite with a hole cut out of it to create, since the source itself is actually a square, it's like 20 centimeters. So we use this mask to create a cylindrical, cylindrical plasma, and it had an E-anode for this plasma. The anode for the main plasma is about 50 centimeters away. So this is net current free, but for creating the magnetic flux rope, we put an anode some 13 meters away in this configuration. So we have a long current here where we can vary the current in the background magnetic field and vary this instability threshold. Some of the plasma parameters are down here. Some of the scale plasma parameters, beta, which is the ratio of the kinetic thermal pressure to the background magnetic field pressure. In this experiment, it's varied from 0.01 to 0.1. In the solar wind, this is about one, so we'd like to push this in that direction. Magnetic Reynolds numbers are very high as well as the Lundquist number. And one final feature of the mask, which produces the cylindrical column is that on the other side of it is a wire structure devised to produce a slight magnetic perturbation. So the kink oscillation will occur sometime during the plasma if it is kink unstable. We don't know exactly when it will happen. If you apply a small seed perturbation, just a few gals out of the hundreds of gals that might be present in the background magnetic field, you can initialize or basically set the initial phase of the kink oscillation. You can use that to perform averaging the data that I'll show next. You don't have to do that. You could also use correlation functions, but we find that you can do either one. Well, I will use a variety of two types of antennas that are inserted in the middle of the plasma, and I will get to those in a little bit. They're basically designed to launch three different polarizations of the wave. And in cylindrical geometry, they'll launch an M equals zero mode of the alphane wave, an M equals minus one mode, or an M equals plus one. So I'm going to have three different source waves. And this is a movie, exactly two movies side by side. They're taking, the machine operates in a pulse mode where the plasma is created once per second. So we, and we keep the machine on for three or four months at a time. This experiment went on for about a week and a half. So each shot, the plasma is very reproducible from shot to shot. And with the seeding of the kink oscillation, these are actually two different discharges. Here we go. So this is the anode in the back. This is one of the antennas, and you see a whole bunch of probes coming in from one side. This view is from one end of the device. So the source for the magnetic flux rope is behind, is out of the page here. And the main plasma source is far off and into the distance. So you can see the oscillation of the kink mode. And again, they are, they're phase locked throughout the experiment. So we can vary, as I said, the current that comes out of the cathode and vary the background magnetic field. So this just to establish whether we are launching or exciting a kink instability. So if we rewrite that condition I provided earlier and very use the magnetic field keeping these other, the current and the current constant, the length is constant, and the size of it is constant. If we just vary the background magnetic field, we can see this would be unstable if the magnetic field satisfies this criteria. And for us, this is 500 gal. So what we did was vary the background magnetic field from 1500 gal down to 350 gal. So these two cases should be unstable to the kink instability and the rest, green through magenta, should be stable. So what we find, this is the driving frequency of the alphane wave and this is the spontaneous that we observed. So this is done with the modulation or seeding off. So this is the primary kink frequency and you can see some harmonics there. And you can see that they occur primarily when this primarily grows when we cross this kink stability threshold. So we're exciting this, this is trying to demonstrate that we're exciting this mode. And you also notice the production of these sidebands around the primary driver frequency. So these dotted waves I'll identify as, and you'll see there are others, but I'm gonna focus on the lower sideband. I'll denote it with an L, the upper sideband with a U. I will call the kink mode, denote that with K, the antenna or alphane wave frequency. I'm gonna label it with an A. So now I'm gonna look at the mode structure. I'll show you the instantaneous mode patterns of the magnetic field in planes perpendicular to the magnetic, the background magnetic field. So I mentioned three types of antennas that we're gonna use. The kink mode itself has a natural m equals minus one character. So with the driver, with three different drivers, in order to satisfy these frequency matching conditions for the three wave process, so these fall right on the difference of the kink frequency. So the frequency matching conditions are easily seen from the power spectrum. But these m number matching conditions should also match for all these three combinations. So we have these three drivers, one kink mode, and an upper and lower sideband. So in total we're gonna have to have match six different m mode patterns. And here they are. So again, this is data taken at a grid of 35 by 35 spatial locations, so that's 12, 25 spatial locations. The probe visits this site, accumulates an ensemble average of 20 shots and then moves to the next location. So these data are acquired by the computer overnight. So this is the time series frequency filtered for the kink mode here for the alphane driver frequency. In this case, the m equals zero mode. And then it's further frequency filtered at the daughter of frequencies, the upper and lower sidebands. And then the mode pattern is reconstructed here. So you can see in this case, the kink mode is fixed, the upper band should have, which is the sum frequency, should have a mode number of zero minus one or minus one in total. The lower band should have minus, minus one or plus one. So we should have for combination of this plus and minus one mode numbers. So you can kind of see the two current channels that are formed here, that's an m equals one structure. If we do a decomposition in Fourier space, and we can see that primarily that the upper mode, which is sorry, the lower mode, which is in blue, the upper is in red. So the lower band indeed has the plus one mode number. The driver is of course fixed at m equals zero. The kink mode is primarily at m equals minus one or there's some other components. And the lower band, lower mode is at minus one. So it satisfies these m number matching conditions. And for the other two cases, what we expect for now we have the same driver pattern for the kink mode, the Alfven driver in the m equals plus one case then should produce an m equals zero and m equals plus two mode pattern. So this is kind of m equals zero pattern and m equals, I don't know what here, but that's what this graph is for. This tells you that primarily this, this is the lower band. So the lower band is in blue. It's primarily in the plus two state so there are other m harmonics. So that's all the six daughter waves that satisfy this as mutable mode number matching condition, if you will. Oh, sorry, I guess I didn't do this one. This is the opposite of the other one except now the lower band is this, we go back, sorry. So the upper band was zero, the bottom one was one. If we switch the polarization of the driver, these two flip and now you get minus two mode and m equals zero and indeed the modem matching number. So that is the sixth total as mutable modem matching conditions that I wanted to show you. Now what about the transfer of energy from the larger to the lower scale? So what we do is again we're in cylindrical geometry so we do a Bessel function decomposition of the radial mode patterns. So for the kink instability and the driver mode, I'm just gonna show the m equals zero mode from now on. The m equals zero mode is a very large spatial pattern so it really hits the limit of the k spectrum that we can measure here but it's close. And I've scaled this to the k perp times the ion gyro radius which is the dissipation scale for the kinetic scales for the ions. This is ultimately one of the places where the energy can wind up in a cascade. So we're not near k perp equals one. The both the kink and the alphane wave have k perp row i close to point one. For both the upper and lower side band, the decomposition yields the k perp row i closer to point three for both of them. So you can see roughly a doubling or a doubling in k space of where the power is for these lower modes. Although the total amplitude as you saw from the power spectrum is not as high as the driver mode so we're not pumping an equal amount there but we are transferring energy from these larger to lower scales, smaller scales. Okay now is this really a quadratic interaction between the two? So I'm gonna turn to the bispectral analysis to demonstrate that this is actually happening. So for those of you not familiar with the bispectrum, it's a third order spectral quantity. The brackets denote an ensemble average over n realizations. I'm gonna show you some data where the number of realizations is about 1500. Each of these is a Fourier transform of the time series. So from a single probe. So this and the third unit here. So you have one single time series. You compare, you notice this is a function of two different frequencies. It's one frequency you're interested in, a second frequency you're interested in and a third frequency which matches the three wave coupling. So if you then weight this by the power that's in the cross spectrum and the auto-spectra for the third wave and weight that by the absolute value squared of the bispectrum, you get the bi-coherence. So if the waves which satisfy this three wave condition are in phase at all for the entire spectrum for the entire ensemble, then this value will go to one. So this, and if they're completely out of phase then by the time we're done with the ensemble averaging, this will take this to be zero. So this is a value that goes from zero to one and it gives you directly the amount of power that is transferred to the amount of power that is in a particular mode which is due to the other two frequencies. And here's the data. So this is the bi-coherence and the bi-coherence is normally shown in a very large plane but because of the wide separation in frequencies and the fast digitization rate, you wouldn't be able to see many of these little dots. So I'm just gonna show you. The region of interest of the bispectrum near the alphane driver frequency and the kink frequencies and its harmonics. So you can see in particular, I'm gonna like to focus on just two of these. So this is at the frequency FK and FA. So this dot indicates the amount of power from the alphane wave to and the kink wave at the upper frequency. So this is F-alphane plus F-kink. So this shows a 40% of the power that is in that mode came from a quadratic interaction between the two other modes, the two drivers. So the value is not as large. This is at a frequency which is down by one kink frequency from the driver mode of the alphane wave and then up along the vertical axis back to the kink mode. This only has somewhere around 20% or less of the power. Seems to have come from a quadratic interaction between the two. Do you also notice some of these other modes which are the lower side bands? I told you in the beginning, it was only gonna talk about the primary daughter waves but you saw those other side bands that look like higher order daughter waves. So there seems, those apparently have also a similar energy content or power which is due to the interaction between the daughter and those side bands and you can also see it starts interacting with multiples of the kink wave frequency. And if I would show the entire spectrum that you can also see quadratic interactions of this with itself which leads to higher order harmonics of quadratic interaction of this with itself and a higher quadratic interaction of this with itself but this is what I wanted to focus on now just to demonstrate where this energy is coming from. Okay, so I think I've been on time. So in summary, we've made kink unstable flux ropes in the laboratory. In this case, we were able to see the phase of the kink oscillation so that we could do some averaging but that's not a necessary condition for anything that I showed you here. We launched the shear waves with n equals zero and plus one m number as milithum modes and watched the interaction of those with the kink mode. We observed side bands of the launch frequency. We identified them in the power spectrum. They easily satisfied their frequency, three wave matching criterion. I do say the phase locking allows us to do the measurements. Basically, you can average it rather than doing the cross-correlation. So in a sense, it's just the analysis that's a little simpler due to this step. It has muthal wave patterns for all six of the daughter ways. We're in general good agreement between the observed and expected mode numbers. This special function decomposition showed, I showed you them equals zero driver, the transfer of energy from this larger scales down to the smaller scales. And this is by factor of two. And the bispectral analysis showed us that this was a quadratic interaction between the kink wave and the alfane wave. And in particular, maybe 20 to 40% of the power is directly shown to be due to this nonlinear action between the two waves. And that's all I have for you. Thank you.